Optimal. Leaf size=125 \[ \frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{a^3 d \sqrt {a+b}}-\frac {(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\left (3 a^2-4 a b+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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Rubi [A] time = 0.19, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3186, 414, 527, 522, 206, 208} \[ -\frac {\left (3 a^2-4 a b+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{a^3 d \sqrt {a+b}}-\frac {(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 414
Rule 522
Rule 527
Rule 3186
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a+b-b x^2\right )} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\operatorname {Subst}\left (\int \frac {3 a-b-3 b x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\cos (c+d x)\right )}{4 a d}\\ &=-\frac {(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\operatorname {Subst}\left (\int \frac {3 a^2-a b+4 b^2-(3 a-4 b) b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\cos (c+d x)\right )}{8 a^2 d}\\ &=-\frac {(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {\left (3 a^2-4 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 a^3 d}\\ &=-\frac {\left (3 a^2-4 a b+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \cos (c+d x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b} d}-\frac {(3 a-4 b) \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [C] time = 6.30, size = 657, normalized size = 5.26 \[ \frac {b^{5/2} \csc ^2(c+d x) (-2 a+b \cos (2 (c+d x))-b) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {b} \cos \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-a-b}}\right )}{2 a^3 d \sqrt {-a-b} \left (a \csc ^2(c+d x)+b\right )}+\frac {b^{5/2} \csc ^2(c+d x) (-2 a+b \cos (2 (c+d x))-b) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {b} \cos \left (\frac {1}{2} (c+d x)\right )+i \sqrt {a} \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {-a-b}}\right )}{2 a^3 d \sqrt {-a-b} \left (a \csc ^2(c+d x)+b\right )}+\frac {(3 a-4 b) \csc ^2(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (-2 a+b \cos (2 (c+d x))-b)}{64 a^2 d \left (a \csc ^2(c+d x)+b\right )}+\frac {(4 b-3 a) \csc ^2(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (-2 a+b \cos (2 (c+d x))-b)}{64 a^2 d \left (a \csc ^2(c+d x)+b\right )}+\frac {\left (3 a^2-4 a b+8 b^2\right ) \csc ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (-2 a+b \cos (2 (c+d x))-b)}{16 a^3 d \left (a \csc ^2(c+d x)+b\right )}+\frac {\left (-3 a^2+4 a b-8 b^2\right ) \csc ^2(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (-2 a+b \cos (2 (c+d x))-b)}{16 a^3 d \left (a \csc ^2(c+d x)+b\right )}+\frac {\csc ^2(c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) (-2 a+b \cos (2 (c+d x))-b)}{128 a d \left (a \csc ^2(c+d x)+b\right )}-\frac {\csc ^2(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (-2 a+b \cos (2 (c+d x))-b)}{128 a d \left (a \csc ^2(c+d x)+b\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 612, normalized size = 4.90 \[ \left [\frac {2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (b^{2} \cos \left (d x + c\right )^{4} - 2 \, b^{2} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {\frac {b}{a + b}} \log \left (\frac {b \cos \left (d x + c\right )^{2} + 2 \, {\left (a + b\right )} \sqrt {\frac {b}{a + b}} \cos \left (d x + c\right ) + a + b}{b \cos \left (d x + c\right )^{2} - a - b}\right ) - 2 \, {\left (5 \, a^{2} - 4 \, a b\right )} \cos \left (d x + c\right ) - {\left ({\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}}, \frac {2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \cos \left (d x + c\right )^{3} - 16 \, {\left (b^{2} \cos \left (d x + c\right )^{4} - 2 \, b^{2} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {-\frac {b}{a + b}} \arctan \left (\sqrt {-\frac {b}{a + b}} \cos \left (d x + c\right )\right ) - 2 \, {\left (5 \, a^{2} - 4 \, a b\right )} \cos \left (d x + c\right ) - {\left ({\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 334, normalized size = 2.67 \[ -\frac {\frac {64 \, b^{3} \arctan \left (\frac {b \cos \left (d x + c\right ) + a + b}{\sqrt {-a b - b^{2}} \cos \left (d x + c\right ) + \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{3}} + \frac {\frac {8 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2}} - \frac {4 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3}} + \frac {{\left (a^{2} - \frac {8 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {24 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 255, normalized size = 2.04 \[ -\frac {1}{16 a d \left (\cos \left (d x +c \right )-1\right )^{2}}+\frac {3}{16 a d \left (\cos \left (d x +c \right )-1\right )}-\frac {b}{4 d \,a^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 \ln \left (\cos \left (d x +c \right )-1\right )}{16 a d}-\frac {\ln \left (\cos \left (d x +c \right )-1\right ) b}{4 d \,a^{2}}+\frac {\ln \left (\cos \left (d x +c \right )-1\right ) b^{2}}{2 d \,a^{3}}+\frac {b^{3} \arctanh \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{d \,a^{3} \sqrt {\left (a +b \right ) b}}+\frac {1}{16 a d \left (1+\cos \left (d x +c \right )\right )^{2}}+\frac {3}{16 a d \left (1+\cos \left (d x +c \right )\right )}-\frac {b}{4 d \,a^{2} \left (1+\cos \left (d x +c \right )\right )}-\frac {3 \ln \left (1+\cos \left (d x +c \right )\right )}{16 a d}+\frac {\ln \left (1+\cos \left (d x +c \right )\right ) b}{4 d \,a^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right ) b^{2}}{2 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 181, normalized size = 1.45 \[ -\frac {\frac {8 \, b^{3} \log \left (\frac {b \cos \left (d x + c\right ) - \sqrt {{\left (a + b\right )} b}}{b \cos \left (d x + c\right ) + \sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a^{3}} - \frac {2 \, {\left ({\left (3 \, a - 4 \, b\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a - 4 \, b\right )} \cos \left (d x + c\right )\right )}}{a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}} + \frac {{\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {{\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.93, size = 1105, normalized size = 8.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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